Lemma 1.2 (Heat kernel bounds, [1]}, [2]}) 
If \(V\in \mathcal {K}(M)\) , then for \(0<t\le 1\) , there is a uniform constant \(c=c_{M,V}>0\)  so that
\(e^{-tH_V}(x,y)\lesssim {\left\lbrace \begin{array}{ll}t^{-n/2}e^{-cd_g(x,y)^2/t},\ \text{if}\ d_g(x,y)\le \text{Inj}(M)/2\\1,\ \ \text{otherwise}.\end{array}\right.}\) 

